Stair-climbing vehicles having a pair of spiders, each provided with radially-disposed wheels are well known. Some of these spider wheels are free to rotate about their individual axes, or are rotationally driven even while the spiders are operative on a staircase. Both constructions are dangerous, since they may cause the vehicle to roll backwards or tip over when on the staircase. Other types of such vehicles make use of endless belts. Such types of vehicles lack traction and are apt to damage the stair edges. They are also excessively complex in construction.